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Arthur CHARPENTIER - tails of Archimedean copulas
Tails of Archimedean Copulas
tail dependence in risk management
Arthur Charpentier
CREM-Universite Rennes 1
(joint work with Johan Segers, UCLN)
http ://perso.univ-rennes1.fr/arthur.charpentier/
Colloque Evaluation et couverture des risques extremes
Universite Paris-Dauphine & Chaire AXA de la Fondation du Risque, Juin 2008
1
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Arthur CHARPENTIER - tails of Archimedean copulas
Tail behavior and risk management
In reinsurance (XS) pricing, use of Pickands-Balkema-de Haan’s theorem
Theorem 1. F ∈MDA (Gξ) if and only if
limu→xF
sup0<x<xF
{∣∣Pr (X − u ≤ x|X > u)−Hξ,σ(u) (≤ x)∣∣} = 0,
for some positive function σ (·), where Hξ,σ (x) =
1− (1 + ξx/σ)−1/ξ , ξ 6= 0
1− exp (−x/σ) , ξ = 0.
1− F (x) ≈ (1− F (u))[1−Hξ,σ(u) (x− u)
], for all x > u.
So, if u = Xk:n, then
1− F (x) ≈ (1− F (Xk:n))︸ ︷︷ ︸≈1−Fn(Xk:n)=k/n
[1−Hξ,σ(Xk:n) (x−Xk:n)
], for all x > Xk:n,
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Arthur CHARPENTIER - tails of Archimedean copulas
Pure premium of XS contract
Recall that πd = E((X − d)+) with d large, thus,
πd =1
P(X > d)
∫ ∞d
1− F (x)dx
≈ k
n
σ
1− ξ
(1 + ξ
d−Xn−k:n
σ
)1− 1ξ
,
i.e.
πd =k
n
σk
1− ξk
(1 + ξk
d−Xn−k:n
σk
)1− 1ξk
(see e.g. Beirlant et al. (2005).
Possible to derive explicit formulas for any tail risk measure (VaR, TVaR...).
3
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Arthur CHARPENTIER - tails of Archimedean copulas
Extending extreme value theory in higher dimension
univariate case bivariate case
limiting distribution dependence structure of
of Xn:n (G.E.V.) componentwise maximum
when n→∞, i.e. Hξ (Xn:n, Yn:n)
(Fisher-Tippet)
dependence structure of
limiting distribution (X,Y ) |X > x, Y > y
of X|X > x (G.P.D.) when x, y →∞when x→∞, i.e. Gξ,σ dependence structure of
(Balkema-de Haan-Pickands) (X,Y ) |X > x
when x→∞
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Arthur CHARPENTIER - tails of Archimedean copulas
Tail dependence in risk management
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1e+01 1e+03 1e+05
1e
+0
11
e+
02
1e
+0
31
e+
04
1e
+0
5
Loss (log scale)
Allo
cate
d E
xpe
nse
s
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1 10 100 1000 100001
e+
00
1e
+0
21
e+
04
1e
+0
6
Car claims (log scale)
Ho
use
ho
ld c
laim
s
Fig. 1 – Multiple risks issues.
5
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Arthur CHARPENTIER - tails of Archimedean copulas
Motivations : dependence and copulas
Definition 2. A copula C is a joint distribution function on [0, 1]d, withuniform margins on [0, 1].
Theorem 3. (Sklar) Let C be a copula, and F1, . . . , Fd be d marginaldistributions, then F (x) = C(F1(x1), . . . , Fd(xd)) is a distribution function, withF ∈ F(F1, . . . , Fd).
Conversely, if F ∈ F(F1, . . . , Fd), there exists C such thatF (x) = C(F1(x1), . . . , Fd(xd)). Further, if the Fi’s are continuous, then C isunique, and given by
C(u) = F (F−11 (u1), . . . , F−1
d (ud)) for all ui ∈ [0, 1]
We will then define the copula of F , or the copula of X.
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Arthur CHARPENTIER - tails of Archimedean copulas
XY
Z
Fonction de répartition à marges uniformes
Fig. 2 – Graphical representation of a copula, C(u, v) = P(U ≤ u, V ≤ v).
7
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Arthur CHARPENTIER - tails of Archimedean copulas
xx
z
Densité d’une loi à marges uniformes
Fig. 3 – Density of a copula, c(u, v) =∂2C(u, v)∂u∂v
.
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Arthur CHARPENTIER - tails of Archimedean copulas
Strong tail dependence
Joe (1993) defined, in the bivariate case a tail dependence measure.
Definition 4. Let (X,Y ) denote a random pair, the upper and lower taildependence parameters are defined, if the limit exist, as
λL = limu→0
P(X ≤ F−1
X (u) |Y ≤ F−1Y (u)
),
= limu→0
P (U ≤ u|V ≤ u) = limu→0
C(u, u)u
,
and
λU = limu→1
P(X > F−1
X (u) |Y > F−1Y (u)
)= lim
u→0P (U > 1− u|V ≤ 1− u) = lim
u→0
C?(u, u)u
.
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Arthur CHARPENTIER - tails of Archimedean copulas
Gaussian copula
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
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0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
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1.0
L and R concentration functions
L function (lower tails) R function (upper tails)
GAUSSIAN
●
●
Fig. 4 – L and R cumulative curves.
10
![Page 11: Slides dauphine](https://reader035.fdocuments.us/reader035/viewer/2022062319/55853584d8b42a9b388b5261/html5/thumbnails/11.jpg)
Arthur CHARPENTIER - tails of Archimedean copulas
Gumbel copula
0.0 0.2 0.4 0.6 0.8 1.0
0.0
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0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
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1.0
L and R concentration functions
L function (lower tails) R function (upper tails)
GUMBEL
●
●
Fig. 5 – L and R cumulative curves.
11
![Page 12: Slides dauphine](https://reader035.fdocuments.us/reader035/viewer/2022062319/55853584d8b42a9b388b5261/html5/thumbnails/12.jpg)
Arthur CHARPENTIER - tails of Archimedean copulas
Clayton copula
0.0 0.2 0.4 0.6 0.8 1.0
0.0
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0.0 0.2 0.4 0.6 0.8 1.0
0.0
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1.0
L and R concentration functions
L function (lower tails) R function (upper tails)
CLAYTON
●
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Fig. 6 – L and R cumulative curves.
12
![Page 13: Slides dauphine](https://reader035.fdocuments.us/reader035/viewer/2022062319/55853584d8b42a9b388b5261/html5/thumbnails/13.jpg)
Arthur CHARPENTIER - tails of Archimedean copulas
Student t copula
0.0 0.2 0.4 0.6 0.8 1.0
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0.0 0.2 0.4 0.6 0.8 1.0
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L and R concentration functions
L function (lower tails) R function (upper tails)
STUDENT (df=5)
●
●
Fig. 7 – L and R cumulative curves.
13
![Page 14: Slides dauphine](https://reader035.fdocuments.us/reader035/viewer/2022062319/55853584d8b42a9b388b5261/html5/thumbnails/14.jpg)
Arthur CHARPENTIER - tails of Archimedean copulas
Student t copula
0.0 0.2 0.4 0.6 0.8 1.0
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0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
L and R concentration functions
L function (lower tails) R function (upper tails)
STUDENT (df=3)
●
●
Fig. 8 – L and R cumulative curves.
14
![Page 15: Slides dauphine](https://reader035.fdocuments.us/reader035/viewer/2022062319/55853584d8b42a9b388b5261/html5/thumbnails/15.jpg)
Arthur CHARPENTIER - tails of Archimedean copulas
Weak tail dependence
If X and Y are independent (in tails), for u large enough
P(X > F−1X (u), Y > F−1
Y (u)) = P(X > F−1X (u)) · P(Y > F−1
Y (u)) = (1− u)2,
or equivalently, log P(X > F−1X (u), Y > F−1
Y (u)) = 2 · log(1− u). Further, if Xand Y are comonotonic (in tails), for u large enough
P(X > F−1X (u), Y > F−1
Y (u)) = P(X > F−1X (u)) = (1− u)1,
or equivalently, log P(X > F−1X (u), Y > F−1
Y (u)) = 1 · log(1− u).
=⇒ limit of the ratiolog(1− u)
log P(Z1 > F−11 (u), Z2 > F−1
2 (u)).
15
![Page 16: Slides dauphine](https://reader035.fdocuments.us/reader035/viewer/2022062319/55853584d8b42a9b388b5261/html5/thumbnails/16.jpg)
Arthur CHARPENTIER - tails of Archimedean copulas
Weak tail dependence
Coles, Heffernan & Tawn (1999) defined
Definition 5. Let (X,Y ) denote a random pair, the upper and lower taildependence parameters are defined, if the limit exist, as
ηL = limu→0
log(u)log P(Z1 ≤ F−1
1 (u), Z2 ≤ F−12 (u))
= limu→0
log(u)logC(u, u)
,
and
ηU = limu→1
log(1− u)log P(Z1 > F−1
1 (u), Z2 > F−12 (u))
= limu→0
log(u)logC?(u, u)
.
16
![Page 17: Slides dauphine](https://reader035.fdocuments.us/reader035/viewer/2022062319/55853584d8b42a9b388b5261/html5/thumbnails/17.jpg)
Arthur CHARPENTIER - tails of Archimedean copulas
Gaussian copula
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
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0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Chi dependence functions
lower tails upper tails
GAUSSIAN
●●
Fig. 9 – χ functions.
17
![Page 18: Slides dauphine](https://reader035.fdocuments.us/reader035/viewer/2022062319/55853584d8b42a9b388b5261/html5/thumbnails/18.jpg)
Arthur CHARPENTIER - tails of Archimedean copulas
Gumbel copula
0.0 0.2 0.4 0.6 0.8 1.0
0.0
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0.0 0.2 0.4 0.6 0.8 1.0
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Chi dependence functions
lower tails upper tails
GUMBEL
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Fig. 10 – χ functions.
18
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Arthur CHARPENTIER - tails of Archimedean copulas
Clayton copula
0.0 0.2 0.4 0.6 0.8 1.0
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0.0 0.2 0.4 0.6 0.8 1.0
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Chi dependence functions
lower tails upper tails
CLAYTON
●
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Fig. 11 – χ functions.
19
![Page 20: Slides dauphine](https://reader035.fdocuments.us/reader035/viewer/2022062319/55853584d8b42a9b388b5261/html5/thumbnails/20.jpg)
Arthur CHARPENTIER - tails of Archimedean copulas
Student t copula
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
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0.0 0.2 0.4 0.6 0.8 1.0
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Chi dependence functions
lower tails upper tails
STUDENT (df=3)
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Fig. 12 – χ functions.
20
![Page 21: Slides dauphine](https://reader035.fdocuments.us/reader035/viewer/2022062319/55853584d8b42a9b388b5261/html5/thumbnails/21.jpg)
Arthur CHARPENTIER - tails of Archimedean copulas
Application in risk management : Loss-ALAE
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0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Loss
Allo
cate
d E
xpe
nse
s
Fig. 13 – Losses and allocated expenses.
21
![Page 22: Slides dauphine](https://reader035.fdocuments.us/reader035/viewer/2022062319/55853584d8b42a9b388b5261/html5/thumbnails/22.jpg)
Arthur CHARPENTIER - tails of Archimedean copulas
Application in risk management : Loss-ALAE
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
L and R concentration functions
L function (lower tails) R function (upper tails)
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Gumbel copula
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0.0 0.2 0.4 0.6 0.8 1.00
.00
.20
.40
.60
.81
.0
Chi dependence functions
lower tails upper tails
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Gumbel copula
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Fig. 14 – L and R cumulative curves, and χ functions.
22
![Page 23: Slides dauphine](https://reader035.fdocuments.us/reader035/viewer/2022062319/55853584d8b42a9b388b5261/html5/thumbnails/23.jpg)
Arthur CHARPENTIER - tails of Archimedean copulas
Application in risk management : car-household
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0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Car claims
Ho
use
ho
ld c
laim
s
Fig. 15 – Motor and Household claims.
23
![Page 24: Slides dauphine](https://reader035.fdocuments.us/reader035/viewer/2022062319/55853584d8b42a9b388b5261/html5/thumbnails/24.jpg)
Arthur CHARPENTIER - tails of Archimedean copulas
Application in risk management : car-household
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
L and R concentration functions
L function (lower tails) R function (upper tails)
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
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Gumbel copula
●
●
0.0 0.2 0.4 0.6 0.8 1.00
.00
.20
.40
.60
.81
.0
Chi dependence functions
lower tails upper tails
●●●●●●●●●●●●●●●●●●●●●
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●
●
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Gumbel copula
●
●
Fig. 16 – L and R cumulative curves, and χ functions.
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Arthur CHARPENTIER - tails of Archimedean copulas
Archimedean copulas
Definition 6. A copula C is called Archimedean if it is of the form
C(u1, · · · , ud) = φ−1 (φ(u1) + · · ·+ φ(ud)) ,
where the generator φ : [0, 1]→ [0,∞] is convex, decreasing and satisfies φ(1) = 0.
A necessary and sufficient condition is that φ−1 is d-monotone.
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Arthur CHARPENTIER - tails of Archimedean copulas
Some examples of Archimedean copulas
φ(t) range θ
(1) 1θ
(t−θ − 1) [−1, 0) ∪ (0,∞) Clayton, Clayton (1978)
(2) (1 − t)θ [1,∞)
(3) log 1−θ(1−t)t
[−1, 1) Ali-Mikhail-Haq
(4) (− log t)θ [1,∞) Gumbel, Gumbel (1960), Hougaard (1986)
(5) − log e−θt−1e−θ−1
(−∞, 0) ∪ (0,∞) Frank, Frank (1979), Nelsen (1987)
(6) − log{1 − (1 − t)θ} [1,∞) Joe, Frank (1981), Joe (1993)
(7) − log{θt + (1 − θ)} (0, 1]
(8) 1−t1+(θ−1)t [1,∞)
(9) log(1 − θ log t) (0, 1] Barnett (1980), Gumbel (1960)
(10) log(2t−θ − 1) (0, 1]
(11) log(2 − tθ) (0, 1/2]
(12) ( 1t− 1)θ [1,∞)
(13) (1 − log t)θ − 1 (0,∞)
(14) (t−1/θ − 1)θ [1,∞)
(15) (1 − t1/θ)θ [1,∞) Genest & Ghoudi (1994)
(16) ( θt
+ 1)(1 − t) [0,∞)
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Arthur CHARPENTIER - tails of Archimedean copulas
Why Archimedean copulas ?
Assume that X and Y are conditionally independent, given the value of anheterogeneous component Θ. Assume further that
P(X ≤ x|Θ = θ) = (GX(x))θ and P(Y ≤ y|Θ = θ) = (GY (y))θ
for some baseline distribution functions GX and GY . Then
F (x, y) = P(X ≤ x, Y ≤ y) = E(P(X ≤ x, Y ≤ y|Θ = θ))
= E(P(X ≤ x|Θ = θ)× P(Y ≤ y|Θ = θ))
= E((GX(x))Θ × (GY (y))Θ
)= ψ(− logGX(x)− logGY (y))
where ψ denotes the Laplace transform of Θ, i.e. ψ(t) = E(e−tΘ). Since
FX(x) = ψ(− logGX(x)) and FY (y) = ψ(− logGY (y))
and thus, the joint distribution of (X,Y ) satisfies
F (x, y) = ψ(ψ−1(FX(x)) + ψ−1(FY (y))).
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Arthur CHARPENTIER - tails of Archimedean copulas
0 5 10 15
05
1015
20
Conditional independence, two classes
!3 !2 !1 0 1 2 3
!3
!2
!1
01
23
Conditional independence, two classes
Fig. 17 – Two classes of risks, (Xi, Yi) and (Φ−1(FX(Xi)),Φ−1(FY (Yi))).
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Arthur CHARPENTIER - tails of Archimedean copulas
0 5 10 15 20 25 30
010
2030
40
Conditional independence, three classes
!3 !2 !1 0 1 2 3
!3
!2
!1
01
23
Conditional independence, three classes
Fig. 18 – Three classes of risks, (Xi, Yi) and (Φ−1(FX(Xi)),Φ−1(FY (Yi))).
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Arthur CHARPENTIER - tails of Archimedean copulas
0 20 40 60 80 100
020
4060
80100
Conditional independence, continuous risk factor
!3 !2 !1 0 1 2 3
!3
!2
!1
01
23
Conditional independence, continuous risk factor
Fig. 19 – Continuous classes of risks, (Xi, Yi) and (Φ−1(FX(Xi)),Φ−1(FY (Yi))).
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Arthur CHARPENTIER - tails of Archimedean copulas
Properties of Archimedean copulas
• the countercomonotonic copula C− is Archimedean, φ(t) = 1− t,• the independent copula C⊥ is Archimedean, φ(t) = − log(t),• the comonotonic copula is not Archimedean (but can be a limit of
Archimedean copulas).
0.2
0.40.6
0.8
u_10.2
0.4
0.6
0.8
u_2
00.
20.
40.
60.
81
Frec
het l
ower
bou
nd
0.2
0.4
0.6
0.8
u_10.2
0.4
0.6
0.8
u_2
00.
20.
40.
60.
81
Inde
pend
ence
cop
ula
0.2
0.40.6
0.8
u_10.2
0.4
0.6
0.8
u_2
00.
20.
40.
60.
81
Frec
het u
pper
bou
nd
0.0 0.2 0.4 0.6 0.8 1.0
0.00.2
0.40.6
0.81.0
Scatterplot, Lower Fréchet!Hoeffding bound
0.0 0.2 0.4 0.6 0.8 1.0
0.00.2
0.40.6
0.81.0
Scatterplot, Indepedent copula random generation
0.0 0.2 0.4 0.6 0.8 1.0
0.00.2
0.40.6
0.81.0
Scatterplot, Upper Fréchet!Hoeffding bound
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Arthur CHARPENTIER - tails of Archimedean copulas
Properties of Archimedean copulas
• Frank copula is the only Archimedean such that (U, V ) L= (1− U, 1− V )(stability by symmetry),
• Gumbel copula is the only Archimedean such that (U, V ) has the same copulaas (max{U1, ..., Un},max{V1, ..., Vn}) for all n ≥ 1 (max-stability),
• Clayton copula is the only Archimedean such that (U, V ) has the same copulaas (U, V ) given (U ≤ u, V ≤ v) (stability by truncature).
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Arthur CHARPENTIER - tails of Archimedean copulas
Lower tails of Archimedean copulas
Study regular variation property of φ at 0,
lims→0
φ(st)φ(s)
= t−θ0 , t ∈ (0,∞)⇐⇒ θ0 = − lims→0
sφ′(s)φ(s)
.
If θ0 > 0 : asymptotic dependence
Proposition 7. If 0 < θ0 <∞, then for every ∅ 6= I ⊂ {1, . . . , d}, every(xi)i∈I ∈ (0,∞)|I| and every (y1, . . . , yd) ∈ (0,∞)d,
lims↓0
Pr[∀i = 1, . . . , d : Ui ≤ syi | ∀i ∈ I : Ui ≤ sxi]
=(∑
i∈Ic y−θ0i +
∑i∈I(xi ∧ yi)−θ0∑
i∈I x−θ0i
)−1/θ0
This is Clayton’s copula.
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Arthur CHARPENTIER - tails of Archimedean copulas
Lower tails of Archimedean copulas
Study regular variation property of φ at 0,
lims→0
φ(st)φ(s)
= t−θ0 , t ∈ (0,∞)⇐⇒ θ0 = − lims→0
sφ′(s)φ(s)
.
If θ0 = 0 : asymptotic independence (dependence in independence) for strictgenerators (φ(0) =∞)Proposition 8. If θ0 = 0 and φ(0) =∞, for every ∅ 6= I ⊂ {1, . . . , d}, every(xi)i∈I ∈ (0,∞)|I| and every (y1, . . . , yd) ∈ (0,∞)d,
lims↓0
Pr[∀i ∈ I : Ui ≤ syi;∀i ∈ Ic : Ui ≤ χs(yi) | ∀i ∈ I : Ui ≤ sxi]
=∏i∈I
(yjxj∧ 1)|I|−κ ∏
i∈Icexp
(−|I|−κy−1
i
),
where χs(·) = φ−1 (−sφ′(s)/·), and κ is the index of regular variation of ψ, withψ(·) = −φ−1(·)φ′(φ−1(·)).
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Arthur CHARPENTIER - tails of Archimedean copulas
Upper tails of Archimedean copulas
Study regular variation property of φ at 1,
lims→0
φ(1− st)φ(1− s)
= tθ1 , t ∈ (1,∞)⇐⇒ θ1 = − lims→0
sφ′(1− s)φ(1− s)
.
If θ1 > 1 : asymptotic dependence
Proposition 9. If 1 < θ0 <∞, then for every ∅ 6= I ⊂ {1, . . . , d}, every(xi)i∈I ∈ (0,∞)|I| and every (y1, . . . , yd) ∈ (0,∞)d,
lims↓0
Pr[∀i = 1, . . . , d : Ui ≥ 1− syi | ∀i ∈ I : Ui ≥ 1− sxi] =rd(z1, . . . , zd; θ1)r|I|((xi)i∈I ; θ1)
where zi = xi ∧ yi for i ∈ I and zi = yi for i ∈ Ic and
rk(u1, . . . , uk; θ1) =∑
∅ 6=J⊂{1,...,k}
(−1)|J|−1(∑i∈J
uθ1j)1/θ1
for integer k ≥ 1 and (u1, . . . , uk) ∈ (0,∞)k.
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Arthur CHARPENTIER - tails of Archimedean copulas
Upper tails of Archimedean copulas
Study regular variation property of φ at 1,
lims→0
φ(1− st)φ(1− s)
= tθ1 , t ∈ (1,∞)⇐⇒ θ1 = − lims→0
sφ′(1− s)φ(1− s)
.
If θ1 > 1 and φ′(1) < 0 : asymptotic independence, or near independence
Proposition 10. If 1 < θ1 = 1 and φ′(1) < 0, then for all (xi)i∈I ∈ (0,∞)|I| and(y1, . . . , yd) ∈ (0, 1]d ,
lims↓0
Pr[∀i ∈ I : Ui ≥ 1− syi;∀i ∈ Ic : Ui ≤ yi | ∀i ∈ I : Ui ≥ 1− sxi]
=∏i∈I
yj ·(−D)|I|φ−1(
∑i∈Ic φ(yi))
(−D)|I|φ−1(0).
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Arthur CHARPENTIER - tails of Archimedean copulas
Upper tails of Archimedean copulas
If θ > 1 and φ′(1) = 0 : asymptotic independence, dependence in independence
Proposition 11. If 1 < θ1 = 1 and φ′(1) = 0, if I ⊂ {1, . . . , d} and |I| ≥ 2, thenfor every (xi)i∈I ∈ (0,∞)|I| and every (y1, . . . , yd) ∈ (0,∞)d,
lims↓0
Pr[∀i = 1, . . . , d : Ui ≥ 1− syi | ∀i ∈ I : Ui ≥ 1− sxi] =rd(z1, . . . , zd)r|I|((xi)i∈I)
where zi = xi ∧ yi for i ∈ I and zi = yi for i ∈ Ic and
rk(u1, . . . , uk) :=∑
∅ 6=J⊂{1,...,k}
(−1)|J|(∑J
uj) log(∑J
uj)
= (k − 2)!∫ u1
0
· · ·∫ uk
0
(t1 + · · ·+ tk)−(k−1)dt1 · · · dtk
for integer k ≥ 2 and (u1, . . . , uk) ∈ (0,∞)k.
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Arthur CHARPENTIER - tails of Archimedean copulas
Tails of Archimedean copulas
• upper tail : calculate φ′(1) and θ1 = − lims→0
sφ′(1− s)φ(1− s)
,
◦ φ′(1) < 0 : asymptotic independence
◦ φ′(1) = 0 et θ1 = 1 : dependence in independence
◦ φ′(1) = 0 et θ1 > 1 : asymptotic dependence
• lower tail : calculate φ(0) and θ0 = − lims→0
sφ′(s)φ(s)
,
◦ φ(0) <∞ : asymptotic independence
◦ φ(0) =∞ et θ0 = 0 : dependence in independence
◦ φ(0) =∞ et θ0 > 0 : asymptotic dependence
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Arthur CHARPENTIER - tails of Archimedean copulas
upper tail lower tail
φ(t) range θ −φ′(1) θ1 φ(0) θ0 κ
(1) 1θ
(t−θ − 1) [−1,∞) 1 1 1(−θ)∨0 θ ∨ 0 ·
(2) (1 − t)θ [1,∞) 1(θ = 1) θ 1 0 ·
(3) log 1−θ(1−t)t
[−1, 1) 1 − θ 1 ∞ 0 0
(4) (− log t)θ [1,∞) 1(θ = 1) θ ∞ 0 1 − 1θ
(5) − log e−θt−1e−θ−1
θeθ−1
1 ∞ 0 0
(6) − log{1 − (1 − t)θ} [1,∞) 1(θ = 1) θ ∞ 0 0
(7) − log{θt + (1 − θ)} (0, 1] θ 1 − log(1 − θ) 0 ·(8) 1−t
1+(θ−1)t [1,∞) 1θ
1 1 0 ·
(9) log(1 − θ log t) (0, 1] θ 1 ∞ 0 −∞(10) log(2t−θ − 1) (0, 1] 2θ 1 ∞ 0 0
(11) log(2 − tθ) (0, 1/2] θ 1 log 2 0 ·(12) ( 1
t− 1)θ [1,∞) 1(θ = 1) θ ∞ θ ·
(13) (1 − log t)θ − 1 (0,∞) θ 0 ∞ 0 1 − 1θ
(14) (t−1/θ − 1)θ [1,∞) 1(θ = 1) θ ∞ 1 ·(15) (1 − t1/θ)θ [1,∞) 1(θ = 1) θ 1 0 ·(16) ( θ
t+ 1)(1 − t) [0,∞) 1 + θ 1 ∞ 1 ·
(17) − log (1+t)−θ−12−θ−1
θ2(2θ−1)
1 ∞ 0 0
(18) eθ/(t−1) [2,∞) 0 ∞ e−θ 0 ·(19) eθ/t − eθ (0,∞) θeθ 1 ∞ ∞ ·
(20) et−θ− e (0,∞) θe 1 ∞ ∞ ·
(21) 1 − {1 − (1 − t)θ}1/θ [1,∞) 1(θ = 1) θ 1 0 ·(22) arcsin(1 − tθ) (0, 1] θ 1 π/2 0 ·
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Arthur CHARPENTIER - tails of Archimedean copulas
How to extend to more general dependence structures ?
• mixtures of generators, since convex sums of generators defines a generator,• the α− β transformations in Nelsen (1999), i.e.
φα(t) = φ(tα) and φβ(t) = [φ(t)]β , where α ∈ (0, 1) and β ∈ (1,∞).
• other transformations, e.g.◦ exp(αφ(t))− 1, α ∈ (0,∞),◦ φ(1− [1− t]α), α ∈ (1,∞),◦ φ(αt)− φ(α), α ∈ (0, 1),
=⇒ can be related to distortion of Archimedean copulas.
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Arthur CHARPENTIER - tails of Archimedean copulas
upper tail lower tail
φα(t) range α φ′α(1) θ1(α) φα(0) θ0(α) κ(α)
(1) (φ(t))α (1,∞) 0 αθ1 (φ(0))α αθ0κα
+ 1 − 1α
(2) eαφ(t)−1α
(0,∞) αφ′(1) θ1αφ(0)−1
α∗ ∗
(3) φ(tα) (0, 1) αφ′(1) θ1 φ(0) αθ0 κ
(4) φ(1 − (1 − t)α) (1,∞) 0 αθ1 φ(0) θ0 κ
(5) φ(αt) − φ(α) (0, 1) αφ′(α) 1 φ(0) − φ(α) θ0 κ
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